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Properties

Label 2-6e4-9.2-c2-0-27
Degree $2$
Conductor $1296$
Sign $0.996 - 0.0871i$
Analytic cond. $35.3134$
Root an. cond. $5.94251$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.34 − 0.776i)5-s + (6.19 − 10.7i)7-s + (12.7 + 7.34i)11-s + (5.40 + 9.35i)13-s + 28.9i·17-s − 3.60·19-s + (12.7 − 7.34i)23-s + (−11.2 + 19.5i)25-s + (24.3 + 14.0i)29-s + (4 + 6.92i)31-s − 19.2i·35-s + 22.5·37-s + (−21.7 + 12.5i)41-s + (−26.5 + 46.0i)43-s + (14.6 + 8.48i)47-s + ⋯
L(s)  = 1  + (0.268 − 0.155i)5-s + (0.885 − 1.53i)7-s + (1.15 + 0.668i)11-s + (0.415 + 0.719i)13-s + 1.70i·17-s − 0.189·19-s + (0.553 − 0.319i)23-s + (−0.451 + 0.782i)25-s + (0.840 + 0.485i)29-s + (0.129 + 0.223i)31-s − 0.549i·35-s + 0.609·37-s + (−0.531 + 0.306i)41-s + (−0.618 + 1.07i)43-s + (0.312 + 0.180i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0871i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.996 - 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $0.996 - 0.0871i$
Analytic conductor: \(35.3134\)
Root analytic conductor: \(5.94251\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1296} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :1),\ 0.996 - 0.0871i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.605421898\)
\(L(\frac12)\) \(\approx\) \(2.605421898\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-1.34 + 0.776i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (-6.19 + 10.7i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-12.7 - 7.34i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-5.40 - 9.35i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 28.9iT - 289T^{2} \)
19 \( 1 + 3.60T + 361T^{2} \)
23 \( 1 + (-12.7 + 7.34i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-24.3 - 14.0i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 22.5T + 1.36e3T^{2} \)
41 \( 1 + (21.7 - 12.5i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (26.5 - 46.0i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-14.6 - 8.48i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 84.5iT - 2.80e3T^{2} \)
59 \( 1 + (-78.8 + 45.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-6.5 + 11.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (20.5 + 35.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 16.3iT - 5.04e3T^{2} \)
73 \( 1 - 71.5T + 5.32e3T^{2} \)
79 \( 1 + (23.3 - 40.4i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (13.2 + 7.65i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 78.9iT - 7.92e3T^{2} \)
97 \( 1 + (45.5 - 78.9i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.569888983314554766248482933082, −8.562347050098573066730960310978, −7.937529695198057740366208304529, −6.82750224425327957605852742430, −6.48760550412640781903401489203, −5.06052531519791338303499925765, −4.22257797608243612383203948187, −3.68161853251145563280570648252, −1.76440378832706522845086197840, −1.19547331783757989666477646572, 0.911952933556152353326940429159, 2.25669068901246197947585549699, 3.07354869556069022659905132477, 4.41227254141978231466633873672, 5.41588220410948527186987258231, 5.95363278473194566220542216172, 6.91792708910938120540404277740, 8.026379298235067705394138756658, 8.743355931276315921932674098287, 9.229320555740110934788981404290

Graph of the $Z$-function along the critical line